OMG it turns out I really do need to know algebra. Mr. Farnsworth, you were right: I should have paid better attention in math class!

This is not homework, it's just something that has been making me curious and & I want to figure it out. Because I'm a stay-at-home mom and I don't have enough things to think about, apparently.

It's a word problem, so here goes:

I went to the store and bought two large containers of shampoo and conditioner. They are each one liter/33.8 ounces. When I shampoo, I use 2 squirts from the shampoo's nozzle. When I condition, I use 5 squirts from the conditioner's nozzle. I assume each squirt dispenses the same amount of product; I like to use more conditioner than shampoo. So here it is, several months later, and I have run out of conditioner while still having plenty of shampoo left.

So, some questions:
1. How can I mathmatically work out how much more quickly I use up the conditioner than the shampoo?
2. If I wanted to go to the store and buy enough bottles of shampoo and conditioner to make it equal -- that I would eventually finish the last bottles of both at the same time -- how would I do that?

I'm interested in knowing the answers, and getting an idea of how this problem is worked out so I can figure out important stuff like this in the future.

Also, I was good at big picture stuff like physics, but I somehow always got lost with the basics. I still don't know how to do fractions, and I'm rusty with my times table. I plan on re-learning this along with my kid as he grows.

You could just buy 2 bottles of shampoo and 5 bottles of conditioner (or multiples of that.)

I don't think I used any algebra to work that out in my head. Just imagine that you had all those bottles lined up and used one squirt from each, every time you washed and conditioned.posted by needs more cowbell at 8:14 PM on January 14, 2010 [4 favorites]

you use one-and-a-half times more conditioner than shampoo, so two shampoo bottles should last through five conditioner bottles.posted by radiosilents at 8:15 PM on January 14, 2010

1. For every 2 squirts of shampoo you use 5 squirts of conditioner. Therefore for each single squirt of shampoo, you use 5 / 2, or 2.5 squirts of conditioner. I.e. you use the conditioner 2.5 times faster.

2. Since during each shower your goop use is 2 parts shampoo and 5 parts conditioner, you'll need to buy two shampoo bottles and five conditioner bottles in order to run out at the same time.

I'm a little puzzled about why you want to run out of them at the same time. Why not just buy more whenever one is getting low? But that's outside the realms of algebra :)posted by Salvor Hardin at 8:20 PM on January 14, 2010

This is how I would work out the problem:

2 units shampoo : 5 units conditioner
-> least common multiple of 2 and 5 is 10
-> the smallest 'unit' of shampoo & conditioner where they would be used up at the same time is (10/2=) 5 bottles of conditioner and (10/5=) 2 bottles of shampoo.
-> any multiple of the smallest 'unit' will also work: i.e. 10 bottles conditioner & 4 bottles conditioner also work

An alternative way of thinking about the problem:
Get 5 bottles of conditioner, open them all.
Get 2 bottles of shampoo, open them all.

Every time you use them, you squirt conditioner 5 times, 1 from each of the 5 bottles. You squirt shampoo 2 times, 1 from each of the 2 bottles. Each time you shower, each bottle gets squirted once. They will, therefore, all finish at the same time.

Are the bottles the same size? Then everyone's answers above are correct.

But let's say, to make things interesting, that the shampoo bottle holds X fluid ounces and the conditioner bottle holds Y fluid ounces.

Then instead of just getting 2 bottles of shampoo and 5 bottles of conditioner, you'd have to get 2*Y bottles of shampoo and 5*X bottles of conditioner (notice the switch of X and Y here- if your conditioner bottle is huge and so Y is big, you're going to have to get a lot of shampoo to run out of shampoo at the same time you run out of conditioner).

But let's say that either X or Y isn't a whole number. Since you can't go out and buy a fractional bottle of shampoo, you need to multiply your numbers by some common factor Z so that 2*Y*Z and 5*X*Z are both whole numbers.

The brute force approach to finding a Z that makes both 2*Y*Z and 5*X*Z whole numbers is to just start with 1 and multiply by 10 until you've exhausted the number of decimal places that the (presumably 100% accurate) measurement on the more precise bottle uses. Unless the capacity of the bottle is given in terms of an infinite decimal, you're golden!posted by Jpfed at 9:02 PM on January 14, 2010

The problem as you've posed it doesn't need a grasp of algebra so much as proportion (assuming that all squirts are, indeed, equal volumes).

1. How can I mathmatically work out how much more quickly I use up the conditioner than the shampoo?

For every two squirts of shampoo, you use five squirts of conditioner. Another way of saying that is that you're using conditioner 5/2 times as quickly as you're using shampoo. Another way of saying 5/2 is two and a half.

2. If I wanted to go to the store and buy enough bottles of shampoo and conditioner to make it equal -- that I would eventually finish the last bottles of both at the same time -- how would I do that?

When you're comparing usage rates, then provided the units you're using are consistent, it doesn't matter what they are. Two and a half times as many squirts of conditioner as shampoo is also two and a half times as many pints or gallons or cubic miles or bottles of conditioner as shampoo. So you should run out of shampoo and conditioner at the same time any time you buy two and a half times as many bottles of conditioner as shampoo: five conditioner and two shampoo, or ten conditioner and four shampoo, or fifteen conditioner and six shampoo, and so on.

I still don't know how to do fractions, and I'm rusty with my times table. I plan on re-learning this along with my kid as he grows.

Get the times tables down first, just as brute facts, before worrying about what they mean; then work on your fractions. If you know without thinking that three fours are twelve, then when you're trying to work out how many twelfths make a quarter, the answer will make sense instead of being just another disconnected brute fact.

The best way I know to learn the times tables is based on the way I was taught them in Grade 3: chanting with minimal fluff. Two twos are four, three twos are six, four twos are eight, five twos are ten... note: not two times two equals four, three times two equals six, four times two equals eight... because not only do all those extra syllables make the chanting take longer, they make the memorized results slower to retrieve.

Picking a table to work on today and chanting it out loud as you do something physical like walking or jogging or turning a skipping rope will lock it in fairly painlessly.

We chanted everything from once one to twelve twelves, which means that I now have a hundred and forty-four times table results taking up space in my head. I think that's actually excessive. You don't need the one-times table, because the rule that once anything is itself is as quick to apply as a table lookup would be. Similarly, you don't need the ten-times table, because the rule that ten anythings means append a zero is equally quick. You don't really need elevens and twelves either, except as occasionally handy alternatives to a multi-digit multiply. All you really need is table entries for two twos through nine nines, which is only sixty-four brute facts instead of a hundred and forty-four; less than half the mental space.

You can cut that down even further, by not bothering to learn the rules where the second number is smaller than the first one: when you need to know what nine sevens are, you swap them and retrieve seven nines are sixty-three instead. I don't think that cutting down from sixty-four facts to thirty-six is worth slowing down your mental processes by inserting an extra comparison step, but some people prefer it.posted by flabdablet at 10:16 PM on January 14, 2010

If you're using 5 squirts of conditioner for 2 squirts of shampoo, and the bottles are the same size, then you'll use 5 bottles of conditioner in the same time as you use 2 bottles of shampoo. Thus you'll need to buy the bottles in a 5/2 ratio. If you want this in decimal form, take a calculator and punch in 5 divided by 2, and you'll get 2.5, i.e. you're using 2.5 times more conditioner than shampoo.posted by creasy boy at 11:56 PM on January 14, 2010

Actually you're using one and a half times more conditioner than shampoo, which is another way of saying two and a half times as much conditioner as shampoo.

If you were using them up at the same rate, you'd be using no more and no less conditioner than shampoo, which is another way of saying just as much conditioner as shampoo.

Some great answers up this thread. The key takeaway for your current problem is the ratio 5:2--that is, for every five bottles of conditioner you buy, you must buy two bottles of shampoo (assuming the bottles are the same size and the pumps deliver the same volume of product per squirt for each).

flabdablet's multiplication-table-memorization method sounds quite reasonable, but then again, I transferred between two schools in the fall of my third-grade year, and the students in the destination school were in the middle of learning the multiplication tables when I transferred. I took the mandatory multiplication-table test every day from the end of the multiplication-tables section of our course to the end of that year, and never passed it. Some years later, I earned a bachelor's degree in mathematics and computer science, with honors.

Your mileage, and your child's for that matter, may vary.posted by tellumo at 12:21 AM on January 15, 2010

Times tables aren't really that important. As long as you know the 'important' pairs you can usually figure out the rest. For example if you know 4*3 = 12, then you see 4*6 it's easy to see that it's 12*2 = 24, for example.

Anyway, times tables and fractions wouldn't have been needed for this problem. You don't need to do any arithmetic at all. 2 squirts and 5 squirts means you need to buy 2 bottles and 5 bottles or order to get them to match up. What's needed is number sense an idea about how numbers work and how math all fits together. All the arithmetic can be done with a calculator.

I did a little bit of googling and I came across this site which has little audio tutorials about the very basics of algebra. You can use that and you can solve basic algebra problems using wolfram alpha. So for example, if you have a question like

"You need to collect 60 wheels from a junkyard. You can buy either motorcycles or cars, but you have to get a total of 20"?

So let's call 'x' the number of bikes and 'y' the number of cars. Since we get 4 wheels from a car, the number of wheels from cars is 4 times y. Since we get two wheels from a bike the number of wheels from bikes is 2 * x.

Since the total is 60 we get 2 * x + 4 * y = 60.

And since we have 20 total vehicles, we know x + y = 20. So how do you figure this out? You just type "2x + 4y = 60 and x + y = 20" into wolfram alpha: like this and it will tell you the answer: ten cars and ten bikes. No need to do tedious arithmetic like you learned in school. Fractions are similarly easy 23323/05434 + 3243/8923198? The answer is 2738596966/638008657

If you want to do math in this day and age, the important thing is to understand the concepts, and how to use the tools available.posted by delmoi at 4:15 AM on January 15, 2010

What's needed is number sense an idea about how numbers work and how math all fits together.

Number sense is a form of intuition. Intuition works on stuff you don't piece together linearly. That's why I believe rote memorization of times tables is still, even in this day and age, a useful activity. Once you've got that set of basic relationships locked in, it really does make growing your number sense much, much easier.

Getting a set of addition tables (from one and one is two all the way up to nine and nine is eighteen) locked in as well is also worth doing. What you're after is completeness of these, so that they're just there without you having to think about them; if you see a six and a seven, you just know they add up to thirteen and multiply to forty-two without needing to work those facts out.

Sure, you can work out the missing entries if your memorized tables have holes, but doing so is a hell of a lot slower and ties up more working memory. Both these effects make doing multi-digit arithmetic tedious and error-prone, which it truly doesn't have to be if your tables are in good shape. And if you repeatedly experience arithmetic as tedious and error-prone, it's pretty easy to conclude that you're one of those people who is just no good at maths.

It truly saddens me to see checkout operators (young ones, mostly) completely and utterly unable to make basic change without making faces of pain and reaching for something with buttons on it.

Rote memorization is absolutely not all there is to mathematics, of course, but a certain amount of time spent doing it pays off bigtime by giving your intuition something to chew on as you grapple with the conceptual stuff.

It's not just tables, either. Lists of standard algebraic transforms [e.g. a2 - b2 = (a + b)(a - b)] and trig formulae and standard forms of integration are all best learned by rote. And that's something I wish I had understood in high school; I was one of the "smart" ones who preferred to work those things out from first principles every time I needed to use them, but I hit the limits of my capacity to do that in about mid year 12 and did quite poorly on the end-of-year exams. The fact that I was still doing better than classmates who were still working out that eight twelves are ninety-six should have been a Clue. Heigh ho.posted by flabdablet at 5:39 AM on January 15, 2010

From shampoo & conditioner to wolfram alpha and trig formulae. I LOVE ASK.METAFILTER! Thanks to all.posted by BlahLaLa at 9:29 AM on January 15, 2010

On a procedural note, you could get around the equal-squirt assumption by measuring how much shampoo you actually have left now that the conditioner is gone. (If the bottle's translucent, just use a ruler to estimate the fraction of empty space.) If you've used, say, a third of the shampoo, then you would need to buy 3 bottles of conditioner for every bottle of shampoo. Or if you've used a quarter of the shampoo, you'd need 4 bottles of conditioner for every bottle of shampoo.posted by stargazer360 at 10:04 AM on January 15, 2010

Getting a set of addition tables (from one and one is two all the way up to nine and nine is eighteen) locked in as well is also worth doing. What you're after is completeness of these, so that they're just there without you having to think about them; if you see a six and a seven, you just know they add up to thirteen and multiply to forty-two without needing to work those facts out.

Sure, you can work out the missing entries if your memorized tables have holes, but doing so is a hell of a lot slower and ties up more working memory. Both these effects make doing multi-digit arithmetic tedious and error-prone, which it truly doesn't have to be if your tables are in good shape. And if you repeatedly experience arithmetic as tedious and error-prone, it's pretty easy to conclude that you're one of those people who is just no good at maths.

Sure but if a person finds rote memorization of times tables tedious and boring it's easy to conclude they're someone who "hates math" Since it's not necessary it's not a good idea to try to memorize it unless you actually like doing it.posted by delmoi at 1:26 PM on January 15, 2010

Well, none of it's really necessary, is it?

Rote memorization is really not so hard. What makes it tedious and difficult is (a) trying to do too much of it at once (b) being forced to do it without any kind of rationale, as opposed to deliberately deciding to invest some time in it (c) trying to do it quietly (d) trying to do it while sitting still (e) expecting it to be tedious, so that instead of simply doing it, you waste a bunch of time reminding yourself how bored you are; this tends to create tables peppered with useless entries like "five sevens are BORING".

The actual memorization part isn't inherently any harder than remembering any of the stuff that gets pounded into your head by advertisers or TV or Top 40 radio. Repetition is the key. You don't need to try to memorize this stuff (in fact, doing so is counter-productive; once again, you tend to end up with "three eights are I CAN"T REMEMBER THIS SHIT". All you need to do is pick one table row per day - say, four twos up to four nines - and say it over and over whenever you're doing something rhythmic.

Rote learning is the mental equivalent of bicep curls. No, it's not necessary, but it's very, very helpful; and just like physical exercise, it's most helpful if done voluntarily and often, with more attention paid to form than outcome.posted by flabdablet at 1:48 PM on January 15, 2010

Well, none of it's really necessary, is it?

The poster wants the skills to apply mathematical thinking to everyday problems. To extend your analogy about exercise, not everyone does the same ones. If you want to run marathons, you don't do a lot of arm curls, compared to someone who wants to do rock climbing.

If the poster wants to learn math, she should focus on what interests here. Some people find geometry interesting, some people find calculus interesting and some people might enjoy arithmetic. But computers have freed us from needing to be really good at arithmetic in order to enjoy and appreciate the more interesting forms of mathematics. You still need to have a good sense for how addition and multiplication work, but the times table isn't needed.posted by delmoi at 5:26 PM on January 15, 2010

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